This section is intended to introduce various aspects of the art, which may be associated with exemplary embodiments of the present technological advantage. This discussion is believed to assist in providing a framework to facilitate a better understanding of particular aspects of the present technological advancement. Accordingly, it should be understood that this section should be read in this light, and not necessarily as admissions of prior art.
An important goal of seismic prospecting is to accurately image subsurface structures commonly referred to as reflectors. Seismic prospecting is facilitated by obtaining raw seismic data during performance of a seismic survey. During a seismic survey, seismic energy can be generated at ground or sea level by, for example, a controlled explosion (or other form of source, such as vibrators), and delivered to the earth. Seismic waves are reflected from underground structures and are received by a number of sensors/receivers, such as geophones. The seismic data received by the geophones is processed in an effort to create an accurate mapping of the underground environment. The processed data is then examined with a goal of identifying geological formations that may contain hydrocarbons (e.g., oil and/or natural gas).
Full wavefield inversion is a nonlinear inversion technique that recovers the earth model by minimizing the mismatch between the simulated and the observed seismic wavefields. The Earth model comprises multiple parameters: P-wave velocity, anisotropy, attenuation, etc. In particular, seismic attenuation describes the fact that Earth's interior dissipates and disperses propagating seismic waves. The attenuation effects are particularly prominent in the presence of gas. If attenuation was not taken into account in FWI, the inverted P-wave velocity will be lower than it should be (e.g., Cheng et al., 2015). Furthermore, if unreliable velocity models are used for seismic depth imaging, the structures beneath gas bodies would be distorted. Therefore, attenuation plays an important role in correct estimation of P-wave velocity. P-waves, or compressional waves, are waves in which the vibrations occur in the direction of propagation of the waves.
Both P-wave velocity (Vp) and P-wave attenuation quality factor (Qp) have impact on the travel time of the propagating waves. Thus, there is significant ambiguity between the two parameters. Conventional FWI results suffer from crosstalk: the estimated Vp or Qp is a combination of the true Vp and true Qp.
Seismic waves attenuate for a variety of reasons as they travel in a subsurface environment. A quality metric (sometimes referred to a quality factor) Q is typically used to represent attenuation characteristics of underground formations. In general, Q is inversely proportional to seismic signal attenuation and may range from a value of zero to infinity. More specifically, Q is a dimensionless quality factor that is a ratio of the peak energy of a wave to the dissipated energy. As waves travel, they lose energy with distance and time due to absorption. Such energy loss must be accounted for when restoring seismic amplitudes to perform fluid and lithologic interpretations, such as amplitude versus offset (AVO) analysis. Structures with a relatively high Q value tend to transmit seismic waves with little attenuation. Structures that tend to attenuate seismic energy to a greater degree have lower Q values.
As a multi-parameter optimization problem, FWI of Vp and Qp can be solved using the first-order or second-order optimization methods. On the one hand, the first-order optimization methods, such as steepest descent or nonlinear conjugate gradient, utilize the gradient information of the objective function to define a search direction. Bai et al. (2013) developed cascaded single parameter inversions using the first-order optimization method, i.e., inverting for Qp with fixed Vp and then inverting for Vp with fixed Qp. When the starting models are far away from the true models, cascaded inversions may fail to converge because of the strong dependency between Vp and Qp. Cheng et al. (2015) performed FWI for Vp using the first-order optimization method with fixed background Qp, then derived locally anomalous Qp from the inverted Vp model. The Qp model heavily relies on the assumption on the correlation between Vp and Qp. The efficient line search methods developed by Tang and Ayeni (2015) can be used to estimate the relative scales between the Vp and Qp updates. However, the computational cost is higher compared to the method of the present technological advancement.
On the other hand, second-order methods utilize both the gradient and curvature information of the objective function to determine an optimal search direction (Malinowski et al., 2011; Prieux et al. 2012; Denli et al., 2013). The curvature information is valuable for multi-parameter FWI because it balances the gradient of different parameter classes (e.g., Vp and Qp) and provides meaningful updates for parameter classes with different data sensitivity. The computation associated with the curvature information is however very expensive, which is the main obstacle that prevents the second-order method from being widely used in practice.
A widely used local optimization technique is the gradient-based first-order method, (e.g., steepest descent or nonlinear conjugate gradient), which utilizes only the gradient information of the objective function to define a search direction. Although a gradient-only first-order method is relatively efficient—it requires computing only the gradient of the objective function—its convergence is generally slow. The convergence of FWI can be improved significantly by using a second-order method. This improved convergence is achieved because second-order methods utilize both the gradient and curvature information of the objective function to determine an optimal search direction in model parameter space. (The search direction unit vector s is related to the model update process by mupdated=m+αs, where α (a scalar) is the step size.)
The major difference between first and second order methods is that second-order methods precondition the gradient with the inverse Hessian (e.g., Gauss-Newton/Newton method), or with the inverse of a projected Hessian (e.g., subspace method). The Hessian is a matrix of second-order partial derivatives of the objective function with respect to the model parameters. In general, second-order methods are attractive not only because of their relative fast convergence rate, but also because of the capability to balance the gradients of different parameter classes and provide meaningful updates for parameter classes with different data sensitivities (e.g., velocity, anisotropy, attenuation, etc.) in the context of multi-parameter inversion. In second-order methods, optimum scaling of parameter classes using the Hessian is crucial in multi-parameter inversion, if such parameter classes are to be simultaneously inverted. However, because it is very expensive to compute the inverse of the Hessian, this is a major obstacle for wide adoption of second-order methods in practice. Another disadvantage of second-order methods is that if the objective function is not quadratic or convex (e.g., where the initial model is far from the true model), the Hessian or its approximation may not accurately predict the shape of the objective function. Hence, the gradients for different parameter classes may not be properly scaled, thereby resulting in suboptimal search directions.